Linear Regression Analysis. Analysis of paired data and using a given value of one variable to predict the value of the other

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1 Liear Regressio Aalysis Aalysis of paired data ad usig a give value of oe variable to predict the value of the other

2 Liear Regressio Aalysis E: The chirp rate of crickets is strogly correlated with the outside temperature. O 8 differet days a statisticia wet outside ad measured two quatities: the umber of chirps a cricket made i a miute () ad the outside temperature (y). The data is i the table below. X Chirps i 1 miute Y Outside Temperature ( F) a) Fid the liear correlatio coefficiet r b) Fid the equatio of the least squares regressio lie c) Predict the outside temperature if you hear a cricket chirpig 14 times per miute d) Fid a 95% predictio iterval for the outside temperature if a cricket is chirpig 14 times per miute

3 Temperature (y) Liear Regressio Aalysis E (cotiued): X Chirps i 1 miute Y Outside Temperature ( F) Predictor Variable: X = Number of chirps i 1 miute Respose Variable: Y = Outside Temperature Scatter Plot Crickets Temp vs. chirps a) r =.8693 b) y = # of chirps ()

4 Temperature (y) Liear Regressio Aalysis E (cotiued): X Chirps i 1 miute Y Outside Temperature ( F) Crickets Temp vs. chirps a) r =.8693 b) y = # of chirps () c) If a cricket is chirpig at 14 chirps per miute, the outside temperature is about o y =.536(14) = 11 F

5 Temperature (y) Liear Regressio Aalysis E (cotiued): X Chirps i 1 miute Y Outside Temperature ( F) Crickets Temp vs. chirps a) r = # of chirps () b) y = c) If = 14 chirps / mi, o y = 11 F d) For a 95% predictio iterval, the margi or error E is E = 15.9 F ad the predictio iterval is o o E y E or 85.1 F y F

6 The Liear Correlatio Coefficiet r

7 The Liear Correlatio Coefficiet r r is a umber that ca be calculated from paired data that tells you how close to a straight lie the graph of the data is r is always betwee -1 ad 1 Whe r is 1 or -1, the data form a perfect straight lie Whe r is close to 1 or -1, the graph of the data looks almost like a straight lie, but with a little scatter The further away r is from -1 ad 1 (i.e. closer to ), the less the data looks like a straight lie. It ca look scattered all over the place, or ca form a eat shape (but ot a lie) If r is positive, the data treds upward. That meas: whe icreases, y icreases the slope of the lie is positive If r is egative, the data treds dowward. That meas: whe icreases, y decreases the slope of the lie is egative

8 The Liear Correlatio Coefficiet r r is positive whe, y r = r is positive whe, y 6 4 r is close to 1 (actual r =.9698)

9 The Liear Correlatio Coefficiet r r is egative r = -1 whe, y r is egative whe, y 1 5 r is close to -1 (actual r = -.775)

10 The Liear Correlatio Coefficiet r r is close to whe, y??? (actual r = -.11) r is close to whe, y??? (actual r =.1183)

11 The formula for r is: The Liear Correlatio Coefficiet r r y y y y (more o this later )

12 The Least-Squares Regressio Lie

13 The Least-Squares Regressio Lie Goal: We wat to fid a lie that is as close as possible to all the data poits Every lie has a total error ad we are lookig for the lie with the smallest total error E: Data y Let s see which lie is closer to this data. Our choices i this eample are 13 or

14 E (cotiued): Data 13 The Least-Squares Regressio Lie y What is the total error? E E E? 1 3 No because we do t wat egative umbers to cacel out positive oes E y [3] [ (1) 13] 8 1 E y [13] [ (3) 13] 6 E y [9] [ (4) 13] 4 3 E E E? 1 3 No because absolute values are hard to deal with i math So we ll use E E E 1 3

15 E (cotiued): Data 13 The Least-Squares Regressio Lie y E y [3] [ (1) 13] 8 1 E y [13] [ (3) 13] 6 E y [9] [ (4) 13] 4 3 Total error = E E E ( 8) (6) (4) E y [3] [(1) ] 1 1 E y [13] [(3) ] 5 E y [9] [(4) ] 1 3 Total error = E E E ( 1) (5) ( 1) 1 3 7

16 E (cotiued): Data 13 The Least-Squares Regressio Lie y Total error = 116 Total error = 7 So the lie is closer to the data tha the lie 13 Is there a eve closer lie???

17 The Least-Squares Regressio Lie Usig calculus, it ca be show that give some data, there is a lie that is closer to the data tha ay other lie. This lie is called the Least-Squares Regressio Lie (LSRL) Formula for the Least-Squares Regressio Lie b 1 b b1 y y b y y (more o this later )

18 Predictio Itervals

19 Predictio Itervals The Least-Squares Regressio Lie ca be used to predict the value of y give the value of. The predicted value of y is called The problem: is a poit estimate because it is a 1 umber guess of the value of y. Istead, we wat a iterval estimate (a cofidece iterval). Whe fidig cofidece itervals for problems ivolvig paired data, they are called Predictio Itervals. As usual, to fid a predictio iterval, we eed the margi or error formula E, ad E y E

20 Summary of Formulas Correlatio Coefficiet r: y y y y r LSRL: 1 ˆ b b y 1 y y b y y b Predictio Iterval (use t distributio): 1 y b y b y s e df / 1 1 s t E e E y y E y ˆ ˆ

21 A Detailed Liear Regressio Problem

22 A Detailed Liear Regressio Problem E: I order to study global warmig, the data below was take over differet years (CO levels i parts per millio ad temperature i C) CO Temperature y a) Fid the liear correlatio coefficiet r b) Fid the equatio of the least squares regressio lie c) Predict the temperature of the Earth if the CO levels reach 4 parts per millio d) Costruct a 9% predictio iterval for the temperature of the Earth if the CO levels reach 4 parts per millio

23 A Detailed Liear Regressio Problem E (cotiued): CO Temperature y Before we do ay of the parts of this problem, we eed to fid y y y because these appear i may of the formulas.

24 A Detailed Liear Regressio Problem E (cotiued): CO Temperature y y y y ( 314)(13.9) (317)(14) (369)(14.4)

25 Temperature (Celsius) A Detailed Liear Regressio Problem E (cotiued): CO Temperature y Before we calculate r, here is a graph of the data Temp vs. CO Levels CO Level (ppm) What do you thik r is? r is positive whe, y r is close to 1

26 Temperature (Celsius) A Detailed Liear Regressio Problem E (cotiued): CO Temperature y Temp vs. CO Levels CO Level (ppm) a) To calculate r, plug ito the formula for r r.89 r y y y y (1)( ) (1)( ) (3377) (3377)(141.7) 1(8.39) (141.7)

27 b A Detailed Liear Regressio Problem E (cotiued): CO Temperature y b) To fid the equatio of the regressio lie, plug ito the equatios for b ad b 1 y y (1)( ) (3377)(141.7) b 1 (1)( ) (3377).19 y y (141.7)( ) (3377)( ) (1)( ) (3377) b 1 b so

28 Temperature (Celsius) A Detailed Liear Regressio Problem E (cotiued): CO Temperature y a) r =.89 b) y ˆ Here s what s goig o Temp vs. CO Levels CO Level (ppm)

29 A Detailed Liear Regressio Problem E (cotiued): CO Temperature y c) The whole poit of the regressio lie is to make predictios. To make the predictio, just plug i the give value of to predict the y value Whe the Earth s CO level reaches 4 ppm (that is called umber poit estimate predictio for y is ), the best oe.19(4) o C

30 A Detailed Liear Regressio Problem E (cotiued): CO Temperature y a) r =.89 b) y c) ˆ d) As usual, to fid a predictio iterval, we eed to fid E. To fid E we eed to fid,, ad plug everythig ito the formula for E. t / se 1 cof. level /.1/. 5 t / 1.86 (from table) o C df 1 8

31 A Detailed Liear Regressio Problem E (cotiued): CO Temperature y a) r =.89 b) y c) d) t 1. 86? / ˆ s E =? e o C s e y b 1 y b 8.39 (1.483)(141.7) (.19)( ) y

32 A Detailed Liear Regressio Problem E (cotiued): CO Temperature y a) r =.89 b) y c) d) / ˆ t.347 E =? se o C E t / s e 1 1 (1.86)(.347) ( ) 1( ) (3377).97

33 A Detailed Liear Regressio Problem E (cotiued): CO Temperature y a) r =.89 b) y c) ˆ d) t s. 347 E. 97 / e o C ad the 9% predictio iterval is E y E y o C y o C

34 Temperature (Celsius) A Detailed Liear Regressio Problem E (cotiued): CO Temperature y a) r =.89 b) y c) o o d) C y C ˆ Temp vs. CO Levels o C CO Level (ppm)

35 Homework

36 Homework Sec. 4.1: # s 1-43 odd Sec. 4.: # s 1-31 odd Sec. 14.: # s 1-17 all

37 Thaks everyoe, it s bee a fu semester

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